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Casimir operator in su(2) 
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#1
Mar110, 08:06 PM

P: 5

Hi all,
If I define T_{ij} = a^{+}_{i} a_{j}, then C_{2} = T_{11}T_{11} + T_{12}T_{21} + T_{21}T_{12} + T_{22}T_{22} is a second order casimir operator. For SU(2), it's [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 1) But as I calculate it directly, C_{2} = a^{+}_{1} a_{1}a^{+}_{1} a_{1} + a^{+}_{1} a_{2}a^{+}_{2} a_{1} + a^{+}_{2} a_{1}a^{+}_{1} a_{1} + a^{+}_{2} a_{2}a^{+}_{2} a_{2} = a^{+}_{1} a_{1}a^{+}_{1} a_{1} + a^{+}_{1} (a^{+}_{2}a_{2} + 1)a_{1} + a^{+}_{2} (a^{+}_{1}a_{1} + 1)a_{2} + a^{+}_{2} a_{2}a^{+}_{2} a_{2} = N_{1}N_{1} + N_{1}(N_{2} + 1) + N_{2}(N_{1} + 1) + N_{2}N_{2} = (N_{1} + N_{2})^{2} + N_{1} + N_{2} = N(N + 1) which is different from above. Can you let me know what is wrong with my argument? Thank you very much! 


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